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MMSS
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海洋大學力學聲響振動實驗室MSV LAB HRE NTOU
國立台灣海洋大學河海工程學系陳正宗 教授
河 海 工 程 概 論
結 構 工 程
國立台灣海洋大學河海工程學系
Analysis of acoustic eigenfrequencies and eigenmodes by using t
he meshless method
指導教授 : 陳正宗 教授學 生 : 張 銘 翰
15, June, 2001
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海洋大學力學聲響振動實驗室MSV LAB HRE NTOU
Outlines
I. Numerical Methods.
II. The developments of meshless methods and radial basis functions.
III. The approaching methods of the diagonal elements.
IV. The technique for extracting true eigenvalues.
V. The techniques for filtering out the spurious eigenvalues
VI. Conclusions.
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海洋大學力學聲響振動實驗室MSV LAB HRE NTOU
Numerical Methods Numerical Methods
Mesh MethodsMesh Methods
Finite Difference Method
Finite Difference Method
Meshless Method Meshless Method
Finite Element Method
Finite Element Method
Boundary Element Method
Boundary Element Method
Numerical methods
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海洋大學力學聲響振動實驗室MSV LAB HRE NTOU
Meshless methodsMeshless methods
Variational methods(Galerkin methods)
Variational methods(Galerkin methods) BIEMBIEM BEMBEM
Continuous moving least square
Belyschko et al. (1994)
Continuous moving least square
Belyschko et al. (1994)
Continuous KernelMonagh (1982)Liu et al. (1995)
Continuous KernelMonagh (1982)Liu et al. (1995)
Moving least squareLancaster & Salkauskas Belyschko et al. (1994)Nayroles et al. (1992)
Moving least squareLancaster & Salkauskas Belyschko et al. (1994)Nayroles et al. (1992)
Discrete Kernel
Monaghan (1982)
Liu et al. (1995)
Discrete Kernel
Monaghan (1982)
Liu et al. (1995)
Partitious of UnityBabuska and Melenk
Duarte and Oden (1995)
Partitious of UnityBabuska and Melenk
Duarte and Oden (1995)
Boundary node method
Mukherjee, Huang,
Chen & Kang
Boundary node method
Mukherjee, Huang,
Chen & Kang
Local BIE
(unsymmetric)
Atluri, Zhu, and Sladek
Local BIE
(unsymmetric)
Atluri, Zhu, and Sladek
Local Petrov-Galerkin a
pproach
(symmetric)
Atluri, Zhu, and Liu
Local Petrov-Galerkin a
pproach
(symmetric)
Atluri, Zhu, and Liu
Complete solution+
Particular solution
Complete solution+
Particular solution
Completesolution
Completesolution
Particularsolution
Particularsolution
Volumepotential
Volumepotential
RBFsolution
RBFsolution
Chen & KangChen & Kang
The developments of meshless methods
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海洋大學力學聲響振動實驗室MSV LAB HRE NTOU
The developments of radial basis functions
Radial basis function (RBFs)
Radial basis function (RBFs)
Mesh methodMesh method
Globally-supported RBFsGlobally-supported RBFs
DRBEMNardiniBrebbia
DRBEMNardiniBrebbia
Method of particular integral
Ahmad & Banerjee
Method of particular integral
Ahmad & Banerjee
rr 1)( rCr )(
Meshless methodMeshless method
Globally-supported RBFsGlobally-supported RBFsCompactly-supported RBFs
Wu 、 Buhmann & Wendland
Compactly-supported RBFs
Wu 、 Buhmann & Wendland
Volume potential
Volume potential
Method of fundamental
solution
Method of fundamental
solution
)322581()1()(
)35183()1()(
ln26
1
15
163
3
16
6
19
15
1)(
ln23
4
3
1)(
)530
7282366()1()(
)312164()1()(
328
26
26
5432
232
54
326
324
rrrrr
rrrr
rrr
rrrrr
rrrrr
rr
rrrrr
rrrrr
)(
)(
),( )(
rU
sxU
xsUr
c
c
c
(Potential theory)
This thesisThis thesis
r
kr
xsUrD
krJ
xsUrDk
kr
xsUrD
c
c
c
)sin(
)},( Im{)(3
)(
)},( Im{)(22
)cos(
)},( Im{)( 1
0
:
:
:
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海洋大學力學聲響振動實驗室MSV LAB HRE NTOU
Complex-valued BEMComplex-valued BEM
Half-effort computation
Half-effort computation
Imaginary-partformulations
Imaginary-partformulations
Multiple Reciprocity Method(MRM)
Multiple Reciprocity Method(MRM)
Real-part BEMReal-part BEM
Differed by a complementary solution
Half-effort computation
Singular integralsSingular integrals Avoid singular integralsAvoid singular integrals
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海洋大學力學聲響振動實驗室MSV LAB HRE NTOU
Governing Eq.: Governing Eq.: 0)()( 22 xuk
Two-dimension
Two-dimension
One-dimension
One-dimension
Three-dimension
Three-dimension
)Im(r
eikr
r
kr)sin(
)2
)(Im( 0 krikH
)2
Im(k
ieikr
k
kr
2
)cos()(0 krJ
Imaginary–part formulation
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海洋大學力學聲響振動實驗室MSV LAB HRE NTOU
Double-layer
potential approach
Single-layer
Potential approach
* NDIF method by Kang is the special case
Dirichlet problem
Neumann problem
Dirichlet problem
Neumann problem
Distributed-type
B
I sdBsxsUxu )()(),()(
B
I sdBsxsLxt )()(),()(
B
I sdBsxsTxu )()(),()(
B
I sdBsxsMxt )()(),()(
The distribute and concentrate-type
Dirichlet problem
Neumann problem
Dirichlet problem
Neumann problem
Concentrated-type*
jijj
jijI
i ASMAxsUxu )(),()(
jijj
jijI
i ASMAxsLxt )(),()(
jijj
jijI
i BSMBxsTxu )(),()(
jijj
jijI
i BSMBxsMxt )(),()(
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海洋大學力學聲響振動實驗室MSV LAB HRE NTOU
Influence matricesInfluence matrices
L’Hôpital’s ruleL’Hôpital’s rule
2),(lim
0),(lim
0),(lim
1),(lim
2kxsM
xsL
xsT
xsU
xs
xs
xs
xs
Invariant methodInvariant method
Indeterminate forms ( )Indeterminate forms ( )0
0
The derivation of indeterminate forms
RBF for RBF for )(0 krJ
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海洋大學力學聲響振動實驗室MSV LAB HRE NTOU
)],([ xsU
.)()(22
. ),1( ,,1 ,0 ),()(2
0
kJkJNNa
NNkJkNJ
.)(2)(1 220
m
m kJkJ Addition theorem ofBessel function
)]),(([ xsUdiag .10 a
The diagonal elements of U kernel
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海洋大學力學聲響振動實驗室MSV LAB HRE NTOU
)],([ xsT
.)()( 22
. ),1( ,,1 ,0 ),()(2
0
kJkJkNNb
NNkJkJN
.)()(2)()(0 00
m
mm kJkJkJkJ Addition theorem ofBessel function
)]),(([ xsTdiag .00 b
The diagonal elements of T kernel
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海洋大學力學聲響振動實驗室MSV LAB HRE NTOU
Two-dimensional cavity
Influence matricesU(s,x), L(s,x), T(s,x), M(s,x)
+Indirect method
Influence matricesU(s,x), L(s,x), T(s,x), M(s,x)
+Indirect method
RBF for RBF for )(0 krJ
EigenvaluesEigenvalues
EigenmodesEigenmodes
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海洋大學力學聲響振動實驗室MSV LAB HRE NTOU
Single-layerpotential approach
Single-layerpotential approach
0 2 4 6
k
-40
-30
-20
-10
0
Log
T : T ru e e ig en v a lu eS : S p u rio u s e ig en v a lu e(): A n a ly tica l so lu tio n
T
T T T
K 1
2.405
J0(1)
(2 .405)
K 2
3.830
J1(1)
(3 .832)
K 3
5.130
J2(1)
(5 .136) K 4
5.520
J0(2)
(5 .520)
N u m b er o f n o d es : 1 0k : 0 .0 0 5
T
K 5
6.215
J3(1)
(6 .380)
0 2 4 6
k
-30
-20
-10
0
10
Log
T : T ru e e ig en v a lu eS : S p u rio u s e ig en v a lu e(): A n a ly tica l so lu tio n
ST S
K3.055
J'2(1)
(3 .054)
K 2
3.830
J1(1)
(3 .832)
K4.205
J'3(1)
(4 .201)
K 4
5.375
J0(2)
(5 .520)
K 3
5.135
J2(1)
(5 .136)
K 5
6.295
J3(1)
(6 .380)
N u m b er o f n o d es : 1 0k : 0 .0 0 5
T
K 1
2.405
J0(1)
(2 .405)
S ST T S
T
Eigenfrequencies for Dirichlet B.C.
Double-layerpotential approach
Double-layerpotential approach
True eigenvaluesSpurious eigenvalues
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海洋大學力學聲響振動實驗室MSV LAB HRE NTOU
0 2 4 6
k
-50
-40
-30
-20
-10
0
10
Log
T : T ru e e ig en v a lu eS : S p u rio u s e ig en v a lu e(): A n a ly tica l so lu tio n
TS
T
K 1
3.055
J'2(1)
(3 .054)
K 3
4.205
J'3(1)
(4 .201)
K5.135
J 2(1)
(5 .136)
K 5
6.295
J'5(1)
(6 .416)
K 4
5.340
J'4(1)
(5 .318)
S
K 2
3.830
J'0(1)
(3 .832)
TT
K2.405
J0(1)
(2 .404)
S S S T
N u m b er o f n o d es : 1 0k : 0 .0 0 5
0 2 4 6
k
-50
-40
-30
-20
-10
0
10
Log
T : T ru e e ig en v a lu eS : S p u rio u s e ig en v a lu e(): A n a ly tica l so lu tio n
TT
T T
K 1
1.840
J'1(1)
(1 .841)
K 2
3.055
J'2(1)
(3 .054)
K 3
3.830
J'0(1)
(3 .831)
K 4
4.180
J'3(1)
(4 .201)
T
K 5
5.330
J'1(2)
(5 .331)
T
K 6
6.415
J'5(1)
(6 .415)
N u m b er o f n o d es : 1 0k : 0 .0 0 5
Eigenfrequencies for Neumann B.C.
Single-layerpotential approach
Single-layerpotential approach
Double-layerpotential approach
Double-layerpotential approach
True eigenvaluesSpurious eigenvalues
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海洋大學力學聲響振動實驗室MSV LAB HRE NTOU
To extract the true eigenvalues
Deficient constraintsDeficient constraints
Spurious eigensolutionsSpurious eigensolutions
SVDupdating terms
SVDupdating terms
To extract the true eigensolutionsTo extract the true eigensolutions
Additional constraints
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海洋大學力學聲響振動實驗室MSV LAB HRE NTOU
Direct method for Dirichlet B. C. :
SVD updating terms
SVD updating terms
SVD updating terms
Singular equation(UT method)
Hypersingular equation(LM method)
}{~
jt
0}{
jE
E
L
U
MMSS
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海洋大學力學聲響振動實驗室MSV LAB HRE NTOU
SVD updating terms
SVD updating terms
To extract the true eigenfrequencies
}0{}{ }0{}{}}{{ U
jUjψψi
TUj
Uj
j
Uj
ijjUj
}0{}{ }0{}{}}{{ L
jLjψi
TLj
Lj
j
Lj
ijjLj
0 Li
Ui
}0{}]{[
}0{}]{[
iE
iE
L
U
MMSS
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海洋大學力學聲響振動實驗室MSV LAB HRE NTOU
The degenerate kernels for interior and exterior problems:
,))(cos()()(
2),,(
,))(cos()()(2
),,(),(
RmkRJkJRU
RmkJkRJRUxsU
nmm
E
nmm
I
;
;
The degenerate kernels
X
Y
R
S
X
Interior problem
X
Y
R
S
X
Exterior problem
Blue: field pointsRed: source points
IE UU
MMSS
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海洋大學力學聲響振動實驗室MSV LAB HRE NTOU
I
I
TL
UU
E
E
T
U
L
UIE
IE
IE TL
The degenerate kernels
,))(cos()()(
2),,(
,))(cos()()(2
),,(),(
RmkJkRJk
RT
RmkRJkJk
RTxsT
nmm
E
nmm
I
;
;
,))(cos()()(
2),,(
,))(cos()()(2
),,(),(
RmkJkRJk
RL
RmkRJkJk
RLxsL
nmm
E
nmm
I
;
;
MMSS
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海洋大學力學聲響振動實驗室MSV LAB HRE NTOU
Indirect method for Dirichlet B. C. :
SVD updating terms
SVD updating terms
0}{
jI
I
T
U
}0{}{ }0{}{}}{{ U
jUjψψi
TUj
Uj
j
Uj
ijjUj
}0{}{ }0{}{}}{{ T
jTjψi
TTj
Tj
j
Tj
ijjTj
Singular-layer approach:
Double-layer approach:
sum) no ( ,0)()( ikJkJ ii sum) no ( ,0)()( ikJkJ ii
To extract the true eigenvalues by the indirect method
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海洋大學力學聲響振動實驗室MSV LAB HRE NTOU
Double-layerpotential approach
SVDupdating-terms
0 2 4 6
k
-30
-20
-10
0
10
Log
T : T ru e e ig en v a lu eS : S p u rio u s e ig en v a lu e(): A n a ly tica l so lu tio n
ST S
K3.055
J'2(1)
(3 .054)
K 2
3.830
J1(1)
(3 .832)
K4.205
J'3(1)
(4 .201)
K 4
5.375
J0(2)
(5 .520)
K 3
5.135
J2(1)
(5 .136)
K 5
6.295
J3(1)
(6 .380)
N u m b er o f n o d es : 1 0k : 0 .0 0 5
T
K 1
2.405
J0(1)
(2 .405)
S ST T S
T
Examples for the SVD updating termsFor Dirichlet B.C.
0 2 4 6
k
-40
-30
-20
-10
0
10
Log
T
K 1
2.405
T
K 2
3.832
J1(1)
(3 .832)
T
J0(1)
(2 .405)
K 3
5.136
J2(1)
(5 .136)
K 4
5.520
J0(2)
(5 .520)
K 5
6.368
J3(1)
(6 .380)
T T
T : T ru e e ig en v a lu eS : S p u rio u s e ig en v a lu e(): A n a ly tica l so lu tio n
N u m b er o f n o d es : 1 2k : 0 .0 0 1
I
I
T
UTrue eigenvalues
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海洋大學力學聲響振動實驗室MSV LAB HRE NTOU
Single-layerpotential approach
SVDupdating-terms
For Neumann B.C.
Examples for the SVD updating terms
0 2 4 6
k
-50
-40
-30
-20
-10
0
10
Log
T : T ru e e ig en v a lu eS : S p u rio u s e ig en v a lu e(): A n a ly tica l so lu tio n
TS
T
K 1
3.055
J'2(1)
(3 .054)
K 3
4.205
J'3(1)
(4 .201)
K5.135
J 2(1)
(5 .136)
K 5
6.295
J'5(1)
(6 .416)
K 4
5.340
J'4(1)
(5 .318)
S
K 2
3.830
J'0(1)
(3 .832)
TT
K2.405
J0(1)
(2 .404)
S S S T
N u m b er o f n o d es : 1 0k : 0 .0 0 5
0 2 4 6
k
-40
-30
-20
-10
0
10
Log
T : T ru e e ig en v a lu eS : S p u rio u s e ig en v a lu e(): A n a ly tica l so lu tio n
T
K 1
3.054
J'2(1)
(3 .054)
K 2
3.832
J'0(1)
(3 .832)
K 3
4.201
J'3(1)
(4 .201)
K 4
5.331
J'1(2)
(5 .331)
T T T
N u m b er o f n o d es : 1 2k : 0 .0 0 1
I
I
M
LTrue eigenvalues
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海洋大學力學聲響振動實驗室MSV LAB HRE NTOU
Deficient constraintsDeficient constraints
Spurious eigensolutionsSpurious eigensolutions
Mathematically explainingMathematically explaining
Fredholm alternative theoremFredholm alternative theorem
Numerical techniqueNumerical technique
SVD updating documentsSVD updating documents
To filter out the spurious solutions
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海洋大學力學聲響振動實驗室MSV LAB HRE NTOU
Fredholm’s alternative theorem:
For solving an algebraic system:
Fredholm alternative theorem
Adjoint homogenous sol.
0}{ * b
bAx
bAx
Alternative theoremAlternative theorem
0 det A
0}{
}{}{*
***
A
bAx
bAx 1 Solutionx
: the transpose conjugate matrix ofA*Areal is if * AAA T
MMSS
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海洋大學力學聲響振動實驗室MSV LAB HRE NTOU
For double-layer potential approach:
}{ )],([)(
}{ )],([)(
xsMxt
xsTxu
0][][}{ or 0}{][
][
MT
M
T T
T
T
SVD updating documents
Ab x
0}{or 0}{ AA TT
MMSS
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海洋大學力學聲響振動實驗室MSV LAB HRE NTOU
SVD updating documents
SVD updating documents
Dirichlet problem:
Neumann problem: sum) no ( ,0)()( ikJkJ ii sum) no ( ,0)()( ikJkJ ii
}0{}{ }0{}{}}{{}{
T
jTji
TTj
Tj
j
Tj
ijjTT
j
}0{}{ }0{}{}}{{}{
M
jMji
TMj
Mj
j
Mj
ijjTM
j
For double-layer potential approach:
SVD updating documents
0 Mi
Ti
Spurious modes
T
MMSS
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海洋大學力學聲響振動實驗室MSV LAB HRE NTOU
Double-layerpotential approach
SVDupdating documents
0 2 4 6
k
-30
-20
-10
0
10
Log
T : T ru e e ig en v a lu eS : S p u rio u s e ig en v a lu e(): A n a ly tica l so lu tio n
ST S
K3.055
J'2(1)
(3 .054)
K 2
3.830
J1(1)
(3 .832)
K4.205
J'3(1)
(4 .201)
K 4
5.375
J0(2)
(5 .520)
K 3
5.135
J2(1)
(5 .136)
K 5
6.295
J3(1)
(6 .380)
N u m b er o f n o d es : 1 0k : 0 .0 0 5
T
K 1
2.405
J0(1)
(2 .405)
S ST T S
T
0 2 4 6
k
-40
-30
-20
-10
0
10
Log
T : T ru e e ig en v a lu eS : S p u rio u s e ig en v a lu e(): A n a ly tica l so lu tio n
SS S S
K 1
3.054
J'2(1)
(3 .054)
K 2
3.832
J'0(1)
(3 .832)
K 3
4.201
J'3(1)
(4 .201)
K 4
5.331
J'1(2)
(5 .331)
N u m b er o f n o d es : 1 2k : 0 .0 0 1
For Dirichlet B.C.
][ II MT
Examples for the SVD updating documents
Spurious eigenvalues
MMSS
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海洋大學力學聲響振動實驗室MSV LAB HRE NTOU
Double-layerpotential approach
SVDupdating documents
For Dirichlet B.C.
Examples for the SVD updating document
0 2 4 6
k
-40
-30
-20
-10
0
Log
S
K 1
2.405
K 2
3.830
J1(1)
(3 .832)
J0(1)
(2 .405)
K 3
5.135
J2(1)
(5 .136) K 4
5.520
J0(2)
(5 .520)
K 5
6.367
J3(1)
(6 .380)
T : T ru e e ig en v a lu eS : S p u rio u s e ig en v a lu e(): A n a ly tica l so lu tio n
S S S S
N u m b er o f n o d es : 1 2k : 0 .0 0 1
0 2 4 6
k
-50
-40
-30
-20
-10
0
10
Log
T : T ru e e ig en v a lu eS : S p u rio u s e ig en v a lu e(): A n a ly tica l so lu tio n
TS
T
K 1
3.055
J'2(1)
(3 .054)
K 3
4.205
J'3(1)
(4 .201)
K5.135
J 2(1)
(5 .136)
K 5
6.295
J'5(1)
(6 .416)
K 4
5.340
J'4(1)
(5 .318)
S
K 2
3.830
J'0(1)
(3 .832)
TT
K2.405
J0(1)
(2 .404)
S S S T
N u m b er o f n o d es : 1 0k : 0 .0 0 5
Spurious eigenvalues][ II LU
MMSS
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海洋大學力學聲響振動實驗室MSV LAB HRE NTOU
The interior modes of a circular cavity
Single-layer protential approach
Single-layer protential approach
Double-layer protential approach
Double-layer protential approach Analytical solutionAnalytical solution
-1.00 -0.80 -0.60 -0.40 -0.20 0.00 0.20 0.40 0.60 0.80 1.00
-0.80
-0.60
-0.40
-0.20
0.00
0.20
0.40
0.60
0.80
-1.00 -0.80 -0.60 -0.40 -0.20 0.00 0.20 0.40 0.60 0.80 1.00
-0.80
-0.60
-0.40
-0.20
0.00
0.20
0.40
0.60
0.80
-0 .80 -0.60 -0.40 -0.20 0.00 0.20 0.40 0.60 0.80
-0.80
-0.60
-0.40
-0.20
0.00
0.20
0.40
0.60
0.80
MMSS
30
VV
海洋大學力學聲響振動實驗室MSV LAB HRE NTOU
Single-layer protential approach
Single-layer protential approach
0 2 4 6 8 10
k
-40
-30
-20
-10
0
Log
T : T ru e e ig en v a lu eS : S p u rio u s e ig en v a lu e(): A n a ly tica l so lu tio n
K 1
4.440
K 2
7.030
K 11
(4 .4429)
K 21
(7 .0248)
K 3
8.860
K 22
(8 .8858)
K 4
9.600
K 31
(9 .9346)
T
T TT
N u m b er o f n o d es : 1 2k : 0 .0 0 1
Double-layer protential approach
Double-layer protential approach
0 2 4 6 8 10
k
-25
-20
-15
-10
-5
0
5
Log
T
K 1
4.440
T
K 2
7.060
T
K 11
(4 .4429)
T
T : T ru e e ig en v a lu eS : S p u rio u s e ig en v a lu e(): A n a ly tica l so lu tio n
K 21
(7 .0248)
K 3
8.610
K 22
(8 .8858)
K 4
9.870
K 31
(9 .934)S
S
S
S
S
N u m b er o f n o d es : 1 2k : 0 .0 0 1
SVD techniques for a square cavity
SVDupdating terms
SVDupdating terms
0 2 4 6 8 10
k
-40
-30
-20
-10
0
10
Log
T
K 1
4.440
T
K 2
6.990
T
K 11
(4 .4429)
T
T : T ru e e ig en v a lu eS : S p u rio u s e ig en v a lu e(): A n a ly tica l so lu tio n
K 21
(7 .0248)
K 3
8.610
K 22
(8 .8858)
K 4
9.960
K 31
(9 .9346)
N u m b er o f n o d es : 1 2k : 0 .0 1
True eigenvalues
0 2 4 6 8 10
k
-30
-20
-10
0
10
Log
S
K 1
3.140
K 2
4.470
K 11
(3 .1416)
T : T ru e e ig en v a lu eS : S p u rio u s e ig en v a lu e(): A n a ly tica l so lu tio n
K 21
(4 .4429)
K 3
6.280 K 22
(6 .2831)
K 4
7.050
K 31
(7 .0248)
K 5
9.070
K 32
(8 .8858)
S
S
SS
N u m b er o f n o d es : 1 2k : 0 .0 1
SVDupdating documents
SVDupdating documents
Spurious eigenvalues
For Dirichlet B.C.
MMSS
31
VV
海洋大學力學聲響振動實驗室MSV LAB HRE NTOU
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.000.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.000.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
Single-layer protential approach
Single-layer protential approach
Double-layer protential approach
Double-layer protential approach Analytical solutionAnalytical solution
The interior modes of square cavity
MMSS
32
VV
海洋大學力學聲響振動實驗室MSV LAB HRE NTOU
Three-dimensional spherical cavity
Indirect methodIndirect method
RBF for RBF for r
kr)sin( Continuous systemContinuous system
Degenerate kernelDegenerate kernel
X
Z
Y
S
X
X
Z
Y
S
X
Blue: source pointsRed: field points
EigenvaluesEigenvalues
EigenmodesEigenmodes
EigensolutionsEigensolutions
MMSS
33
VV
海洋大學力學聲響振動實驗室MSV LAB HRE NTOU
Eigenfrequencies for Dirichlet B.C.
0 2 4 6
k
-25
-20
-15
-10
-5
0
5
Log
T : T ru e e ig en v a lu eS : S p u rio u s e ig en v a lu e(): A n a ly tica l so lu tio n
T
K 2
4.478
j1(1)
(4 .4934)
N u m b er o f n o d es : 2 0
k : 0 .0 0 1
T
K 1
3.141
j1(1)
(3 .1416)
T
K 3
5.871 j2
(1)
(5 .7635)
Single-layerpotential approach
Single-layerpotential approach
Double-layerpotential approach
Double-layerpotential approach
0 2 4 6
k
-25
-20
-15
-10
-5
0
5
Log
T : T ru e e ig en v a lu eS : S p u rio u s e ig en v a lu e(): A n a ly tica l so lu tio n
TT
K 3
5.746
j2(1)
(5 .7635)
N u m b er o f n o d es : 2 0
k : 0 .0 0 1
K 1
3.141 j0
(1)
(3 .1416)
T
K 2
4.489
j1(1)
(4 .4934)
S
T
SS S S
True eigenvalues
Supurious eigenvalues
MMSS
34
VV
海洋大學力學聲響振動實驗室MSV LAB HRE NTOU
0 2 4 6
k
-25
-20
-15
-10
-5
0
5
Log
T : T ru e e ig en v a lu eS : S p u rio u s e ig en v a lu e(): A n a ly tica l so lu tio n
T T
K 3
4.489
j '0(1)
(4 .4934)
K 4
4.519
j '3(1)
(4 .5141)
N u m ber o f n o d es : 2 0
k : 0 .0 0 1
T
K 5
5.611
j '4(1)
(5 .6467)
K 2
3.345
j '2(1)
(3 .3421)
T
T
K 1
2.082
j '1(1)
(2 .0816)
S
S SS
0 2 4 6
k
-20
-15
-10
-5
0
5
Log
T : T ru e e ig en v a lu eS : S p u rio u s e ig en v a lu e(): A n a ly tica l so lu tio n
T
T
K 3
4.494
j '0(1)
(4 .4934)
N u m b er o f n o d es : 2 0
k : 0 .0 0 1
T
K 1
2.082
j '1(1)
(2 .0816)
K 2
3.323
j '2(1)
(3 .3421)
T
K 4
5.552
j '4(1)
(5 .6467)
T
K 5
5.912
j '1(2)
(5 .9404)
Single-layerpotential approach
Single-layerpotential approach
Double-layerpotential approach
Double-layerpotential approach
True eigenvalues
Spurious eigenvalues
Eigenfrequencies for Neumann B.C.
MMSS
35
VV
海洋大學力學聲響振動實驗室MSV LAB HRE NTOU
Single-layer protential approach
Single-layer protential approach
0 2 4 6
k
-25
-20
-15
-10
-5
0
5
Log
T : T ru e e ig en v a lu eS : S p u rio u s e ig en v a lu e(): A n a ly tica l so lu tio n
T T
K 3
4.489
j '0(1)
(4 .4934)
K 4
4.519
j '3(1)
(4 .5141)
N u m ber o f n o d es : 2 0
k : 0 .0 0 1
T
K 5
5.611
j '4(1)
(5 .6467)
K 2
3.345
j '2(1)
(3 .3421)
T
T
K 1
2.082
j '1(1)
(2 .0816)
S
S SS
Double-layer protential approach
Double-layer protential approach
0 2 4 6
k
-20
-15
-10
-5
0
5
Log
T : T ru e e ig en v a lu eS : S p u rio u s e ig en v a lu e(): A n a ly tica l so lu tio n
T
T
K 3
4.494
j '0(1)
(4 .4934)
N u m b er o f n od es : 2 0
k : 0 .0 0 1
T
K 1
2.082
j '1(1)
(2 .0816)
K 2
3.323
j '2(1)
(3 .3421)
T
K 4
5.552
j '4(1)
(5 .6467)
T
K 5
5.912
j '1(2)
(5 .9404)
SVD techniques for a spherical cavity
Neumann B.C.
SVDupdating terms
SVDupdating terms
0 2 4 6
k
-20
-15
-10
-5
0
5
Log
T : T ru e e ig en v a lu eS : S p u rio u s e ig en v a lu e(): A n a ly tica l so lu tio n
T
T
K 3
4.517
j '3(1)
(4.4934)
N u m b er o f n od es : 2 0
k : 0 .0 0 1
T
K 1
2.082
j '1(1)
(2.0816)
K 2
3.344
j '2(1)
(3.3421)
T
K 4
5.623
j '4(1)
(5.6467)
True eigenvalues
SVDupdating documents
SVDupdating documents
0 2 4 6
k
-25
-20
-15
-10
-5
0
5
Log
T : T ru e e ig en v a lu eS : S p u rio u s e ig en v a lu e(): A n a ly tica l so lu tio n
S
K 2
4.482
j1(1)
(4 .4934)
N u m b er o f n o d es : 2 0
k : 0 .0 0 1
S
K 1
3.141
j0(1)
(3 .1416)
S
K 3
5.611
j2(1)
(5 .7635)
Spurious eigenvalues
MMSS
36
VV
海洋大學力學聲響振動實驗室MSV LAB HRE NTOU
n =0, k=0.0
0.00 50.00 100.00 150.00 200.00 250.00 300.00 350.000.00
50.00
100.00
150.00
Analyticalmodes
0.00 50.00 100.00 150.00 200.00 250.00 300.00 350.000.00
50.00
100.00
150.00
Single-layerapproach
0.00 50.00 100.00 150.00 200.00 250.00 300.00 350.000.00
50.00
100.00
150.00
Double-layerapproach
m =0
otherwise,)(cos
0 ,1),,1(
immn ep
mnu
Contours for the boundary modes
MMSS
37
VV
海洋大學力學聲響振動實驗室MSV LAB HRE NTOU
1.8 1.9 2 2.1 2.2
k
- 7
- 6
- 5
- 4
- 3
- 2
- 1
Log
T h e 1 s t m in im u m e ig en v a lu e
T h e 2 n d m in im u m e ig en v a lu eT h e 3 rd m in im u m e ig e n v a lu e
T h e 4 th m in im u m e ig e n v a lu e
ith
minimumeigenvalue
Analytical eigenvalue
Numerical eigenvalue
Test
1st
2.082
2.081 ˇ
2nd 2.079 ˇ
3rd 2.079 ˇ
4th 2.035
Multiplicity of eigenvalues
4th: 2.035
Multiplicity=3
MMSS
38
VV
海洋大學力學聲響振動實驗室MSV LAB HRE NTOU
otherwise,)(cos
0 ,1),,1(
immn ep
mnu
m =0m =-1 m =1
n =0
n =1
Spherical harmonics
P. A. NELSON Y. KAHANA
MMSS
39
VV
海洋大學力學聲響振動實驗室MSV LAB HRE NTOU
0.00 50.00 100.00 150.00 200.00 250.00 300.00 350.000.00
50.00
100.00
150.00
0.00 50.00 100.00 150.00 200.00 250.00 300.00 350.000.00
50.00
100.00
150.00
0.00 50.00 100.00 150.00 200.00 250.00 300.00 350.000.00
50.00
100.00
150.00
Single-layerapproach
n =1, k=2.082
0.00 50.00 100.00 150.00 200.00 250.00 300.00 350.000.00
50.00
100.00
150.00
0.00 50.00 100.00 150.00 200.00 250.00 300.00 350.000.00
50.00
100.00
150.00
0.00 50.00 100.00 150.00 200.00 250.00 300.00 350.000.00
50.00
100.00
150.00
Analyticalmodes
0.00 50.00 100.00 150.00 200.00 250.00 300.00 350.000.00
50.00
100.00
150.00
0.00 50.00 100.00 150.00 200.00 250.00 300.00 350.000.00
50.00
100.00
150.00
0.00 50.00 100.00 150.00 200.00 250.00 300.00 350.000.00
50.00
100.00
150.00
Double-layerapproach
m =-1 m =0 m =-1
Contours of the boundary modes
MMSS
40
VV
海洋大學力學聲響振動實驗室MSV LAB HRE NTOU
I. Only the boundary nodes are required such that the influence matrices can be determinated by the two-point function.
II. The diagonal elements can be derived by invariant method.
III. Based on the imaginary-part formulations, the deficient constraints cause the spurious eigensolutions.
IV. The SVD techniques can solve the true or spurious eigensolutions well.
Conclusions
MMSS
41
VV
海洋大學力學聲響振動實驗室MSV LAB HRE NTOU
• To overcome the ill-posed problem, the generalized singular value decomposition (GSVD) could be applied.
• The extension to several problems, degenerate boundary, crack and solid corner may be considered.
• The meshless method based on the imaginary-part formulation can be extented to structure vibration problems.
Further research
MMSS
42
VV
海洋大學力學聲響振動實驗室MSV LAB HRE NTOU
The EndThank you for your kind attention
感謝委員們辛勤指導
MMSS
43
VV
海洋大學力學聲響振動實驗室MSV LAB HRE NTOU
0)( kJm
0)( kJm
E
E
L
U
I
I
T
U0)( kJm
0)( kJm EE ML II MT
0)( kJm0)( kJm
E
E
M
T
I
I
M
L
0)( kJm
0)( kJm EE TU II LU
Boundary Value Problem
Eigen-solution
Density function True and spurious eigenvalues
Single-layer
potential
Double layer
potentialDirect method Indirect method
Dirichlet
Problem
TrueSVD
updating term
SVD updating
term
Spurious
SVDupdating documen
t
SVDupdating documen
t
Neumann
Problem
TrueSVD
Updatingterm
SVD updating
term
Spurious
SVDupdating documen
t
SVDupdating documen
t
SVD techniques for eigensolutions
MMSS
44
VV
海洋大學力學聲響振動實驗室MSV LAB HRE NTOU
To overcome the ill-posed problems
克 服 病 態 問 題 分 析 步 驟
三 維 球 形 聲場二 維 圓 形 聲 場
病 態 指 標 - 條 件 數 病 態 指 標 - 條 件 數
病 態 問 題
SVD 補 充 行SVD 補 充 列
Tikhonov 正規化法
GSVD
克服病態
Preconditioner
MMSS
45
VV
海洋大學力學聲響振動實驗室MSV LAB HRE NTOU
Distributed & concentrated type
Distributed-type Concentrated-type*
Single-layer
potential approach
Double-layer
potential approach
B
I sdBsxsUxu )()(),()(
B
I sdBsxsTxu )()(),()(
B
I sdBsxsMxt )()(),()(
B
I sdBsxsLxt )()(),()(
Dirichlet problem
Neumann problem
Dirichlet problrm
Dirichlet problrm
Dirichlet problrm
Neumann problrm
Neumann problrm
Neumann problrm
jijjj
ijI
i ASMAxsUxu )(),()(
jijxjj
ijI
i BSMBxsLxt )(),()(
jijsjj
ijI
i ASMAxsTxu )(),()(
jijsxjj
ijI
i BSMBxsMxt )(),()(
* NDIF method by Kang is the special case
MMSS
46
VV
海洋大學力學聲響振動實驗室MSV LAB HRE NTOU
One-dimensional duct ---Dirichlet problem
Single-layerpotential approach
0 4 8 12
k
-24
-20
-16
-12
-8
-4
0
Log
3.145
6.285
9.425
12.565
T : T ru e e ig en v a lu eS : S p u rio u s e ig en v a lu e(): A n a ly tica l so lu tio n
N u m b er o f n o d es : 2k : 0 .0 0 1
T TT T
(3 .1416)
(6.2832)
(9.4248)
(12.5663)
Double-layerpotential approach
0 4 8 12
k
-10
-8
-6
-4
-2
0
Log
3.145
6.285
9.425
12.565
T : T ru e e ig en v a lu eS : S p u rio u s e ig en v a lu e(): A n a ly tica l so lu tio n
N u m b er o f n o d es : 2k : 0 .0 01
(3.1416)
(6.2832)
(9.4248)
(12.5663)
T T TT
MMSS
47
VV
海洋大學力學聲響振動實驗室MSV LAB HRE NTOU
One-dimensional duct ---Dirichlet problem
0 0.2 0.4 0.6 0.8 1
a
0.48
0.52
0.56
0.6
0.64
0.68
u(a )
Mode 1k=3.145
0 0.2 0.4 0.6 0.8 1
a
-0 .4
0
0.4
0.8
1.2
sin(ka)
Mode 1k=3.1416
0 0.2 0.4 0.6 0.8 1
a
-0.2
0
0.2
0.4
0.6
u(a )
Mode 1k=3.145
Single-layerpotential approach
Double-layerpotential approach
Analytical mode
MMSS
48
VV
海洋大學力學聲響振動實驗室MSV LAB HRE NTOU
One-dimensional duct ---
Single-layerpotential approach
Double-layerpotential approach
Neumann problem
0 4 8 12
k
-10
-8
-6
-4
-2
0
Log
3.145
6.285
9.425
12.565
T : T ru e e ig en v a lu eS : S p u rio u s e ig en v a lu e(): A n a ly tica l so lu tio n
N u m b er o f n o d es : 2k : 0 .0 0 1
(3.1416)
(6.2832)
(9.4248)
(12.5663)
T T T T
0 4 8 12
k
-20
-16
-12
-8
-4
0
4
Log
3.145 6.285
9.425
12.565
T : T ru e e ig en v a lu eS : S p u rio u s e ig en v a lu e(): A n a ly tica l so lu tio n
N u m ber o f n o d es : 2k : 0 .0 0 1
(3 .1416) (6.2832)
(9.4248)
(12.5663)
TT
T T
MMSS
49
VV
海洋大學力學聲響振動實驗室MSV LAB HRE NTOU
One-dimensional duct ---Neumann problem
Single-layerpotential approach
Double-layerpotential approach
Analytical mode
0 0.2 0 .4 0 .6 0 .8 1
a
-0 .4
0
0 .4
0 .8
1 .2
u(a )
Mode 1k=3.145
0 0.2 0 .4 0 .6 0 .8 1
a
-1 .5
-1
-0 .5
0
0 .5
1
1.5
u(a )
Mode 1k=3.145
0 0.2 0 .4 0 .6 0 .8 1
a
- 4
- 2
0
2
4
sin(ka)
Mode 1k=3.1416
MMSS
50
VV
海洋大學力學聲響振動實驗室MSV LAB HRE NTOU
張銘翰 口試問題記錄 Analysis of acoustic eigenfrequencies and eigenmodes
by using the meshless method
1. 無網格可否做大空間問題 ?2. 此種方法有何限制 ? 病態問題是否加重 ?3. RBF 取法是否唯一 ?RBF 隱含於 kernel?4. 與李慶鋒有何不同 ?5. 為何取虛部 ? 若 kernel 為非奇異時,則 free term 能產生
嗎 ? 跳 還是 u?6. Eq. (2-37) & Eq. (2-38) 為何為零 ? 7. 將 mode 相差調正 ?8. 模態可否疊加實驗之 data 結果 ?9. GSRBF & CSRBF 何者適合大空間問題 ?
何者適合病態問題 ?10. RBF 的定義問題 ?
top related